Parallel transport map along a curve

Let $E\to M$ be a vector bundle with a connection $\nabla$.

Let $\gamma:[a,b]\to M$ be a smooth curve.

Construction

A section $s$ of $E$ along $\gamma$ (that is, $s(t)\in E_{\gamma(t)}$) is called parallel if it satisfies

This is the parallel section equation along a curve .

For each initial vector $v\in E_{\gamma(a)}$, there exists a unique parallel section $s_v$ along $\gamma$ with $s_v(a)=v$. Define the parallel transport map

This is a linear isomorphism, and it depends smoothly on the curve and on the initial value.

This construction is the vector-bundle version of parallel transport ; when $E$ is an associated bundle of a principal bundle with connection, $P_\gamma$ can be obtained from the unique horizontal lift of the base curve .

Basic properties

• If $\gamma$ is reversed, then $P_{\gamma^{-1}}=(P_\gamma)^{-1}$.
• If $\gamma=\gamma_2\star\gamma_1$ is a concatenation, then $P_{\gamma_2\star\gamma_1} = P_{\gamma_2}\circ P_{\gamma_1},$ as in parallel transport respects concatenation .
• For loops $\gamma$ based at $x$, the endomorphisms $P_\gamma:E_x\to E_x$ generate holonomy; compare constructing a holonomy element from a loop and the holonomy group .

Examples

1. Flat transport in a trivial bundle. On $E=M\times \mathbb R^n$ with the flat connection $\nabla=d$, the parallel equation is $\dot s(t)=0$, so $s(t)$ is constant and $P_\gamma=\mathrm{id}_{\mathbb R^n}$ for every curve.

2. Line bundle with a 1-form potential. On the trivial complex line bundle $M\times\mathbb C$, fix a real 1-form $\alpha$ and define $\nabla=d+i\alpha$. Then the parallel equation along $\gamma$ becomes

   $$\dot s(t) + i\,\alpha(\dot\gamma(t))\,s(t)=0,$$

   with solution

   $$P_\gamma(z) = \exp\!\Big(-i\int_\gamma \alpha\Big)\,z.$$

   In the abelian case this matches the familiar “phase accumulation” picture.

3. Levi-Civita parallel transport on the sphere. For the tangent bundle of the unit sphere $S^2$ with its Levi-Civita connection , parallel transport around a geodesic triangle rotates tangent vectors by an angle equal to the triangle’s area (a constant-curvature instance of Gauss–Bonnet). This gives a concrete way to see nontrivial holonomy on $S^2$.